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Search: id:A096315
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| A096315 |
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Numbers n such that the n-dimensional integer lattice Z^n contains n+1 equidistant points (i.e. the vertices of a regular n-simplex). |
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+0
2
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| 1, 3, 7, 8, 9, 11, 15, 17, 19, 23, 24, 25, 27, 31, 33, 35, 39, 43, 47, 48, 49, 51, 55, 57, 59, 63, 67, 71, 73, 75, 79, 80, 81, 83, 87, 89, 91, 95, 97, 99, 103, 105, 107, 111, 115, 119, 120, 121, 123, 127, 129, 131, 135, 139, 143, 145, 147, 151, 155, 159, 161, 163, 167
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Schoenberg proved that a regular n-simplex can be inscribed in Z^n in the following cases and no others. (1) n is even and n+1 is a square. (2) n = 3 (mod 4). (3) n = 1 (mod 4) and n+1 is the sum of two squares.
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REFERENCES
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I. J. Schoenberg, Regular Simplices and Quadratic Forms, J. London Math. Soc. 12 (1937) 48-55.
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EXAMPLE
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There is no equilateral triangle in the plane whose vertices have integer coordinates, so 2 is not on the list. But there is a regular tetrahedron in space whose vertices have integer coordinates, namely (0,0,0), (0,1,1), (1,0,1), (1,1,0), hence 3 is on the list.
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MAPLE
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select(n->(is(n, even) and issqr(n+1)) or (n mod 4 = 3) or ((n mod 4 = 1) and (numtheory[sum2sqr](n+1)<>[])), [ $1..200]);
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CROSSREFS
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Sequence in context: A116034 A122987 A047530 this_sequence A112680 A096079 A094551
Adjacent sequences: A096312 A096313 A096314 this_sequence A096316 A096317 A096318
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KEYWORD
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easy,nonn
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AUTHOR
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David G. Radcliffe (radcl008(AT)umn.edu), Aug 01 2004
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